Informal seminar series on numerical analysis and applied mathematics aimed at students.
The aim of the meetings is to present numerical analysis research topics in an accessible manner and to involve interested students. The seminars will have an introductory first part and will also be accessible to those unfamiliar with the subject. They will be held mainly in Italian, in line with the informal tone of the series.
Master’s students and Bachelor’s students who are familiar with the contents of the Scientific Computing course are encouraged to attend.
Organised by PhD students from the University of Pisa and the Scuola Normale Superiore.
The Mackey–Glass equation has become an essential reference in the field of delay differential equations (DDEs).From a strictly mathematical point of view, it represents a cornerstone benchmark to understand emerging analytical techniques or to test advanced computational methods. Despite the great attention it has received in almost half a century, a rigorous mathematical proof of chaos in the Mackey-Glass equation is still an open problem. However, several numerical simulations highlight the chaotic nature of the Mackey-Glass system.
Inspired by the pioneering work of Lorenz , in this talk we present several computational results showing the presence of tent-like scalar maps in the dynamics of the Mackey-Glass equation. An in-depth study on tent maps in the dynamics of this system has not been provided yet. The numerical evidence of the presence of such maps can be a strong indicator of chaotic behavior, since their iteration gives rise to chaotic dynamics. Our latest results lead to the following numerical evidence: for certain values of the parameters, initial state and and for a certain interval I, we can find a sequence of critical pints (Mn)n of the corresponding solution that lie in the interval I and such that their return map approximate arbitrairly well the graph of any tent map F : I → I.
Quantum circuits naturally implement unitary operations, whereas many applications require non-unitary transformations. Block encodings enable these transformations by introducing ancilla qubits and using measurements to effectively realize the desired operation. Despite their theoretical relevance, block encodings often remain impractical on current quantum hardware due to their high resource requirements. A standard construction technique is the Linear Combination of Unitaries (LCU) framework, which is general and powerful but typically incurs substantial gate overhead. To address this limitation, we introduce Fast One-Qubit-Controlled Select LCU (FOQCS-LCU), a compact LCU formulation that exploits the structure of the underlying Hamiltonian. By expressing the Hamiltonian as a linear combination of tensor products of Pauli operators, our construction requires a linear number of ancilla qubits and enables a constant-depth implementation of the SELECT oracle using two fully parallelizable layers of singly controlled Pauli gates. We further develop a parametrized family of Dicke state preparation routines for the state preparation oracle, reducing the gate count compared with standard approaches. Building on these techniques, we derive explicit block-encoding circuits for representative physical models, including Heisenberg and Fermi Hubbard Hamiltonians, and provide detailed non-asymptotic resource estimates. Finally, we extend our approach to matrix polynomial transformations through the FOQCS-LCU framework. The resulting circuits can be controlled with negligible overhead, enabling efficient implementations of subroutines such as Hadamard tests. These results establish a practical framework for resource efficient block encodings and their extensions to structured quantum linear algebra.
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